独立邻域过程
The independent neighborhoods process
摘要：
在$r$ -均匀超图中的三角形$T^{(r)}$是一组$r+1$边，使得它们中的$r$共享公共的$(r-1)$ -顶点集，并且最后一条边包含来自每个前$r$边的剩余顶点。我们的主要结果是$n$点上的随机贪婪无三角过程终止于具有独立数$O((n \log n)^{1/r})$的$r$ -一致超图。因此，利用超图中独立集的最新结果，Ramsey数$r(T^{(r)}, K_s^{(r)})$具有数量级$s^r/\log s$。这回答了~ \cite{BFM, KMV}中提出的问题，并将Ajtai-Komlós-Szemerédi~ \cite{AKS}和Kim~ \cite{K}的著名结果推广到超图上.
英文摘要：
A triangle $T^{(r)}$ in an $r$-uniform hypergraph is a set of $r+1$ edges such that $r$ of them share a common $(r-1)$-set of vertices and the last edge contains the remaining vertex from each of the first $r$ edges. Our main result is that the random greedy triangle-free process on $n$ points terminates in an $r$-uniform hypergraph with independence number $O((n \log n)^{1/r})$. As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number $r(T^{(r)}, K_s^{(r)})$ has order of magnitude $s^r/\log s$. This answers questions posed in~\cite{BFM, KMV} and generalizes the celebrated results of Ajtai-Komlós-Szemerédi~\cite{AKS} and Kim~\cite{K} to hypergraphs.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
